Charting the Unknown: First Passage Time Probabilities for Pearson Diffusion Process and Application to Options Risk Management

Abstract

The first passage time probabilities have applications in many fields, including Finance, Marketing, Economics, Physics, and Statistics. In this paper, we study the first passage time probabilities for a Pearson diffusion process and obtain the lower and upper bounds of the first passage time density. We show that the density may be approximated by the upper bound with an error of approximately five percent. We present an application by modelling the profit and loss function of the S&P 500, FTSE 100 and DAX 40 index options using a Pearson diffusion process. Further, we establish the relation between first passage time probabilities and MaxVaR, i.e., the intra-horizon risk and obtain the MaxVaR for various index options based on first passage time probabilities. This is important as MaxVaR can capture the risk and potential losses incurred at any time of the trading horizon. In addition, we conduct a sensitivity analysis on the parameters for the purpose of robustness.